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Introduction to Vectors

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In order to represent physical quantities such as position and momentum in more
than one dimension, we must introduce new mathematical objects called
vectors. Technically speaking, a vector is defined as an element of a
vector space, but since we will only be dealing with very special types of
vector spaces (namely, two- and three-dimensional Euclidean space) we can be
more specific. For our purposes, a vector is either an ordered pair or triplet
of numbers. On a two-dimensional plane, for instance, any point
(a, b)
is a
vector. Graphically, we often represent such a vector by drawing an arrow from
the origin to the point, with the tip of the arrow resting at the point. The
situation for three-dimensional vectors is very much the same, with an ordered
triplet
(a, b, c)
being represented by an arrow from the origin to the
corresponding point in three-dimensional space.

The vector
(a, b)
in the Euclidean plane.

Unlike scalars, which have only a value for magnitude, vectors are often
described as objects that have both magnitude and direction. This can be
seen intuitively from the arrow-like representation of a vector in the plane.
The magnitude of the vector is simply the length of the arrow (i.e. the distance
from the point to the origin), and can be easily computed using the
Pythagorean Theorem. The direction of a
vector in two dimensions can be characterized by a single angle
θ
(see
); the direction of a vector in three dimensions can be
specified using two angles (usually denoted
θ
and
μ
).

While these ideas are perfectly valid in our case (since we're dealing with
vectors in finite-dimensional Euclidean space) it is not a good idea to become
too attached to the notions of "direction" and "magnitude" for vectors. For
instance, in quantum mechanics vectors often come in the
form of functions (for instance, a particle wave function), and in such a case
it doesn't make sense to talk about the "direction" of the vector. We don't
have to worry about these complications for now, though, and in the following
SparkNote we will rely heavily on basic geometric notions when we discuss vector
addition and multiplication.